Are FFNN (MLP) Lipschitz functions?
I'm not an expert in this area, but it would appear to depend on the choice of activation function:


$e^x$ is not Lipschitz continuous. See Analytic functions which are not Lipschitz continuous.
$\tanh(x)$ is.


That said, this paper appears to give some conditions (specifically for dynamic ANNs) for which networks with activation function involving $e^x$ can be Lipschitz continuous, so possibly the above is not the whole story.